Anisotropic micropolar fluids subject to a uniform microtorque: the stable case

TL;DR

This paper proves the nonlinear asymptotic stability of a specific equilibrium in a 3D anisotropic micropolar fluid system under uniform microtorque, using a novel energy method addressing complex dissipative-conservative interactions.

Contribution

It introduces a new energy-based proof for stability of anisotropic micropolar fluids with microstructure, overcoming challenges from weak dissipation and partial control of microinertia components.

Findings

Nonlinear asymptotic stability of equilibrium for inertially oblate microstructure

Development of a combined energy-dissipation estimate method

Existence of global-in-time decaying solutions near equilibrium

Abstract

We study a three-dimensional, incompressible, viscous, micropolar fluid with anisotropic microstructure on a periodic domain. Subject to a uniform microtorque, this system admits a unique nontrivial equilibrium. We prove that when the microstructure is inertially oblate (i.e. pancake-like) this equilibrium is nonlinearly asymptotically stable. Our proof employs a nonlinear energy method built from the natural energy dissipation structure of the problem. Numerous difficulties arise due to the dissipative-conservative structure of the problem. Indeed, the dissipation fails to be coercive over the energy, which itself is weakly coupled in the sense that, while it provides estimates for the fluid velocity and microstructure angular velocity, it only provides control of two of the six components of the microinertia tensor. To overcome these problems, our method relies on a delicate combination of two distinct tiers of energy-dissipation estimates, together with transport-like advection-rotation estimates for the microinertia. When combined with a quantitative rigidity result for the microinertia, these allow us to deduce the existence of global-in-time decaying solutions near equilibrium.